If logea+b2=12logea+logebthen
a=b
2a=b
a=2b
a=b3
Explanation for the correct option:
logea+b2=12logea+logeb
⇒ logea+b2=12logeab [∵logam+logan=logamn]
⇒ logea+b2=logeab12 [∵nlogam=logamn]
⇒ logea+b2=logeab
⇒ a+b2=ab [∵logam=logan⇒m=n]
Squaring on both sides
⇒ a+b24=ab
⇒ a2+2ab+b2=4ab
⇒ a2-2ab+b2=0
⇒ a-b2=0
⇒ a-b=0
⇒ a=b
Hence, option(A) i.e. a=b is the correct answer.
Let [k] denotes the greatest integer less than or equal to k. Then the number of positive integral solutions of the equation [x[π2]]=⎡⎢ ⎢⎣x[1112]⎤⎥ ⎥⎦ is